Convolution

Convolution

In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions and and produces a third function that in a sense represents the amount of overlap between and a reversed and translated version of . A convolution is a kind of very general moving average, as one can see by taking one of the functions to be an indicator function of an interval.

Table of contents

1 Uses
2 Definition
3 Properties

3.1 Commutativity:
3.2 Associativity:
3.3 Distributivity:
3.4 Associativity with scalar multiplication:
3.5 Convolution theorem:
3.6 Convolutions on Groups

Uses

Convolution and related operations are found in many applications of engineering and mathematics.

In statistics, as noted above, a weighted moving average is a convolution.
In statistics, the probably distribution of the sum of two random variables is the convolution of each of their distributions.
In optics, many kinds of "blur" are described by convolutions. A shadow (e.g. the shadow on the table when you hold your hand between the table and a light source) is the convolution of the shape of the light source that is casting the shadow and the object whose shadow is being cast. An out-of-focus photograph is the convolution of the sharp image with the blur circle formed by the iris diaphragm.
In acoustics, an echo is the convolution of the original sound with a function representing the various objects that are reflecting it.
In electrical engineering, the output of a linear system is the convolution of the input with the system's response to an impulse.
In physics, wherever there a linear system with a "superposition" principle, a convolution operation makes an appearance.

Definition

The convolution of and is written . It is defined as the integral of the product of the two functions after one is reversed and shifted.

The integration range depends on the domain on which the functions are defined. In case of a finite integration range, and are often considered as cyclically extended so that the term does not imply a range violation. Of course, extension with zeros is also possible. The former case of periodic domains is sometimes called a cyclic convolution, while the latter case of zero-extended or infinite domains is sometimes called a linear convolution, especially in the discrete case below.

If and are two independent random variables with probability densities and , respectively, then the probability density of the sum is given by the convolution .

For discrete functions, one can use a discrete version of the convolution. It is then given by

When multiplying two polynomials, the coefficients of the product are given by the convolution of the original coefficient sequences, in this sense (using extension with zeros as mentioned above).

Generalizing the above cases, the convolution can be defined for any two square-integrable functions defined on a locally compact topological group. A different generalization is the convolution of distributions.

Properties

The various convolution operators all satisfy the following properties:

Commutativity:

Note: This property would be lost were one function not reversed as described above.

Associativity:

Distributivity:

Associativity with scalar multiplication:

for any real (or complex) number .

Derivation rule:

where Df denotes the derivative of f or, in the discrete case, the difference operator Df(n) = f(n+1) - f(n).

Convolution theorem:

where F f denotes the Fourier transform of f. This theorem also holds for the Laplace transform.

Convolutions on Groups

If G is a suitable group endowed with a measure m (for instance, a locally compact Hausdorff topological group with the Haar measure) and if f and g are real or complex valued m-integrable functions of G, then we can define their convolution by

In this case, it is also possible to give, for instance, a Convolution Theorem, however it is much more difficult to phrase and requires representation theory for these types of groups and the Peter-Weyl theorem of Harmonic analysis. It is very difficult to do these calculations without more structure, and Lie groups turn out to be the setting in which these things are done.